Some central limit analogues for supercritical Galton-Watson processes

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, if X i , i = 1,2,3, … are independent and identically distributed random variables with EX i = μ, varX i = σ2 < ∞ and \(S_n = \sum\nolimits_{l = 1}^n {X_i } \), then the central limit theorem can be written in the form $$\mathop {\lim }\limits_{n \to \infty } P\left({\sigma ^{ - 1} n^{\frac{1} {2}} \left({n^{ - 1} S_n - \mu } \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leq } x} \right) = \Phi \left(x \right) = \left({2\pi } \right)^{ - \frac{1} {2}} \int_{ - \infty }^x {e^{ - \frac{1} {2}u^2 } du}.$$ This provides information on the rate of convergence in the strong law \(n^{-1} S_n \underset{\rightarrow}{\rm a.s.} \mu {\rm as} n \rightarrow \infty\). (“ a.s. ” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.